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With unit error variance, the Gaussian likelihood has log-likelihoodFull column rank gives $\hat{\beta }=(X^{T}X{)}^{-1}{X}^{T}Y$ and the least-squares identityMultiplying the likelihood by ${\prod }_j\varphi ({\beta }_j)$ and absorbing all terms independent of $\beta$ into $C$ gives

Put $z_i=\log Y_i$. The Weighted graph Laplacian $L_J$ of the tree is defined byMultiplication of the Gaussian likelihood by the prior shows that, conditionally on the precision parameter $\tau$,Completing the square therefore givesAs a function of $\tau$, the posterior density isso, in shape-rate notation,The precision matrix $M$ has the sparsity pattern of a tree. A sparse Cholesky decomposition and its triangular solves have $O(n)$ cost and storage on this graph, while the gamma update also costs $O(n)$. Hence each systematic-scan Gibbs sampler iteration costs $O(n)$.